Function generating circuits requiring only linear elements



Feb. 14, 1961 B. E. coWART ET AL 2,971,698

FUNCTION CENERAIINC CIRCUITS REQUIRINC ONLY LINEAR ELEMENTS Filed Maron14, 1955 I 53,561 I ffl-E7; @ILD 2,971,698 Patented Feb. 14, 1961.

FUNCTION GENERATING CIRCUITS REQUIRING ONLY LINEAR ELEMENTS BrooksEhrmon Cowart, Pacoima, Lloyd David Ball, Los Angeles, George BruerCrane, Redondo Beach, and Orin Henry Knowlton, Jr., Los Angeles, Calif.,assignors to Gilfillan Bros. Inc., Los Angeles, Calif., a corporation ofvCalifornia Filed Mar. 14, 1955, Ser. No. 494,178

1 Claim. (Cl. 23S- 197) This invention relates -to function generatingcircuits requiring only linear elements and, more particularly, to suchcircuits for generating time-varying signals representing such functionsas: hyperbolas; parabolas; and ellipses, only linear circuit elementsbeing required, where these elements may Ialso be of a .passive naturesuch as inductors, capacitors, or resistors.

Many types of function generators for providing time varying functionsare presently `available in the art, where nonlinear elements areemployed. For example, diode limiter circuits may be utilized to providea time varying output voltage Xo which varies in accordance with a timevarying input voltage X1, in -a specified manner; the diodecharacteristic being selected to provide the desired functionaltransition frorn the input signal Xl to the output signal Xo. Suitablecircuits of this type, for example, are shown on pages 273 through 278of a bool; entitled Electronic Analog Computers by Korn and Kornpublished in 1952 by the McGraw-Hill Book Compan New York, Toronto andLondon.

Many other nonlinear elements have been employed such as, for example,thyrites' and ferrites. Moreover, vacuum tubes have been utilized toprovide logarithmic functions by biasing the tube at a nonlinearcharacteristic portion which approximates the desired function.

It is apparent that in utilizing nonlinear elements to provide a desiredtime varying function, considerable care must be exercised to obtainspeciic characteristics, As a result the elements specified arenecessarily ycostly since they must conform with particularspecifications.V

Furthermore, such carefully selected nonlinear elements may vary incharacteristics due toy changes in operating conditions such astemperature or pressure. Thus an accurate function generator employingsuch` elements not only has an added cost factor due to the specialcharacteristics required but also must include means for regulating suchoperating conditions as temperature and pressure to within certainprescribed boundaries.

A further limitation inherent in the prior art practice is the fact thatfrequently the nonlinear element required must be` a vacuum `tube orother active element, where the term active is utilized to indicate thatthe element requires activating power in order for it to function. As aAresult such circuits require power to maintain the nonlinear elements inthe proper operating position and are also subject to inaccuracies dueto variation in the power supply utilized,

, Furthermore, it is apparent that such active elements costconsiderablymore than passive elements such as resistors andcapacitors,for the same degree :of accuracy.

The present invention obviates the above and other disadvantagesinherent in the prior art by providing a class of function generatingcircuits utilizing linear elements, which may all be of a passivenature. According to the basic concept of the invention, the particularfunction to be generated is first translated into a sum of vexponentialfunctions where the exponential functions 2 may be delayed in timeaccording to a. predetermined initial condition of the function desired.

A large class of time-varying functions may be simulated in this manner,where the degree of accuracy inherent in the simulation is a function ofthe number of linear elements employed providing a corresponding numberof exponential Variables. The basic technique of the invention, forexample, may be utilized to simulate a.

time-varying hyperbolic function such as: y=c/t. Since the simulatingexponential functions cannot provide an initial condition of y=oo, the`sirmllation process must begin after a predetermined initial timeinterval which may be designated as ti. This hyperbolic function thenmay be approximated with a reasonable accuracy by. the combination oftwo exponential functions as follows:

Aeree-m L Bene-m gli.

von the other hand, has a small negative slope and therefore may be mostaccurately simulated by an exponential having a small exponential factork2.

In this manner any function may be simulated by4 combining a pluralityof exponentials, where the accuracy inherent in the simulation is afunction of the4 number of points that the exponential's are selectedtoV pass through. The second basic step of the invention is t0 select aset of linear elements, which -rnay all bev passive, to providev thebasic exponential functionerequired. Where two exponentials arespeciiied, for example, two circuits including resistance andcapacitance yelements may be utilized or two equivalent circuits in-vcluding resistance and linductance elements.V

After establishing the basic conguration of linear cir-V cuit elementswhich may provide the simulating eX- ponential functions, thetransformation function for theV simulating circuit arrangement isformulated.V This ktransformation or transfer function may be obtainedby wellknown Laplace-transform operational calculus.

The transformed function is then converted intoa time-varying functionwhere the coefhcients and initial conditions speciiyingthe functionremain as unl-:nowns in terms of the linear circuit elements employed;Suitable values for these linear circuit `elements are then determinedby a simultaneous equation solution Vwhere-the coefcients of the desiredsimulating exponential function series are compared with the unknowncoelicients of the time varying function. i

Many other time varying functions may be generated in this manner. Aparabolic function, for example, may be generated througha combinationof increasing exponential circuits where an initial shift in time may beintroduced corresponding to the y=0 portion of the parabola. It shouldalso be understood that although the actual circuit must function in atime domain, the time variable may nevertheless be considered torepresent other independent variables. Thus the function y=]"(x) may berepresented where x is considered to be equivalent to the independentvariable of time.

lAccordingly, it is an object of the present invention to provide aclass of function generating circuits which utilize only linearelements.

Another object of the invention is to provide a linear 'between thehyperbolic obviating the special charac- -teristics -arid accuratelyregulated temperature and pressure conditions.

A further object is to provide. a class'of function generating circuitswhich may utilize passive elements obviating the necessity for activeelements providing the desired function which require power foractivation, the power being necessarily accurately regulated to providethe desired function. l

Still another object of the invention is to provide a function generatorwhich may utilize passive elements avoiding the inaccuracies resulting1n operation where a well regulated power supply is required foractivating active elements.

Yet a further obiec't is to provide an economical .class of functiongenerating circuits where special nonlinear elements are not requirednor particular operating conditions such as temperature or pressure.

A more specific object of the invention is to provide a class offunctiongenerating circuits where a desired function is simulated byexponential functions, each exponential function being provided by acorresponding set of linear elements which may be passive elements suchas inductors, capacitors or resistors. A

'Ihe novel features which are believed to be .characteristic of theinvention, both as to its organization and method of operation, togetherwith further objects and advantages thereof, will be better understoodfrom the following description considered in connection with theaccompanying drawings. -It is to be expressly understood, however, thatthe drawings are for the purpose of illustration and description only,and are not intended as a definition of Vthe limits of the invention.

Fig. l is a block diagram of a basic embodiment of the invention;

Fig. 2 is a schematic diagram of one species of the invention forproviding av signal simulating a rectangular hyperbolic function;

Fig. 2a is a graph indicating the correspondence function to besimulated and the exponential functions utilized according to thepresentinvention; and

Fig. v2b is an equivalent circuit for the embodiment of Fig. 2.

Reference is now made `to Fig. embodiment of the invention isillustrated in block diagram form. As indicated in Fig. l the basicembodiment comprises a plurality of exponential circuits indicated asproviding the functions:

AgkiU-ti); Balou-ti); NekMt-ti) Each of the exponential circuitsreceives certain initial amplitude conditions, reprcsentedabycorresponding signails, and also initial time conditions in order toestablish in the interval 5 seotSO sec.

Since many curve fitting techniques may be utilized to approximate thisfunction in the interval between tL--S and t=50 it is not deemednecessary to show the specific l wherein the basicV 4 manner of derivinga simulating function. A suitable function is found to be:

It will be noted that this function is expressed in a form where theinitial time condition t1=5 is effectively provided by failing toutilize the iirst 5-second portion of the function which starts from anamplitude 65. 'Ibis function may `also be expressed as follows:

where tS for simulation. 'I'lhis function may be achieved by introducingan actual time delay in initiating the operation of the exponentialcircuits.

This exponential simulating function Ymay be provided by the double RCexponential circuit shown in Fig. 2.

Referring now to Fig. 2, it is noted that the embodiment includes afirst RC exponential circuit including a capacitor C1 :and a resistorR1. The capacitor C1 is initially charged to the voltage E appliedthereto through normally closed relay contact Rta 1 forming part of arelay Rta. Y

Ari output signal is derived from the junction of C1 and R1 and -isapplied to a second RC exponential circuit, including resistor R2 andcapacitor C2, having one end connected to resistor R2 and the other endconnected to ground. Capacitor C2 receives an initial charging voltageof 0 volt through a normally closed second contactl Rta 2 `of relay Rta.It will alsov be noted that a second relay Rtx is included, having 4anormally closed contact Rtx 1 completing connection between capacitor C1and ascisse-621+.scie-wm) g resistor R-l in the first exponentialcircuit. When relayv will be understood that the particular potentialsillus-- trated in Fig. 2 lare by no means intended as a limitation ofthe invention.

The operation of the circuit begins at time ta when relay Rta isactuated opening associated contacts Rt 1 and Rtarz. In the particularoperation to be discussed below, it will be assumed that potential E is65 volts and that vat time fa occurs at zero time as indicated in Fig.2a. It will be understood, however, that other suitable times ta may beselected such as ta: l0 seconds 4at which timey the potentiall acrosscapacitor C1 should be selected to be 17.5 volts as indicated in"Fig.2a. 'Ilhe time condition ta then, corresponds to the general conditionti indicated in Fig; l and discussed-above as covering any of thepossible initial time conditions. Y

After signal ta actuates relay Rta, opening the associated contacts,capacitor C1 begins to discharge through the parallel path of theexponential circuit including the resistor. R2 and capacitor C2 coupledin series and through resistor R1. -It will be Vshown in the detaileddiscussion which follows that the values of the components of theseexponential circuits may be selected to simulate the desired hyperbolicfunction with a high degrec of accuracy during a considerable portion ofthe time interval. Thugs, in Fig. 2a, it will be noted that the functiony=/ t is accurately simulatedby the summation of .two exponentials:y=56.l6ert/595 and y=8.84el=/621 in the interval Istarting from tapproximately equal to 8 seconds through the time t=60 seconds.

In view of the continuous computation of the time varying functionaccording the invention, the desired linal time condition may beselected at any point following the lirst point of reasonable accuracy.Thus, relay Rtx may be actuated at any time following to equal toapproxi-V an output signal indicationrfor the value of the function atthat time.

A more detailed explanation of a particular operation of the inventionasemployed in Ian initial velocity computing circuit is found in copendingapplication Variable Range Signal Generating Circuit With Means forComputing Initial Velocity by Lloyd David Ball et al., tiled March 7,1955, Serial No. 492,482, now Patent No. 2,832,537; and VelocityTracking System For Increasing The Range yof Acquisition of. MovingTargets by Ball et al., filed March 7, 1955, Serial No. 492,627. Thepresent invention is shown in means 200 of Fig. 2` in the The Kirchoilaw equations for the equivalent circuit shown in Fig. Zbmay beexpressed as follows:

These equations may then be transformed according to..

EVS 0 Tin) range signal generating circuit application, Serial No.492,482 and in Fig. 6 in means 610 `of the velocity tracking systemcovered in application Serial No. 492,627. In these applications, thepresent invention is employed to compute initial velocity and isactu-ated at time to to generate a hyperbolically varying signal definedas where Ad is the range difference between two points through which atarget passes over the time interval At.

In `employing the present invention in the above particular application,relays Rto are actuated at the time the target whose initial velocity isto be computed passes a first range point. The circuit values of thehyperbolic function generating circuit of the invention are selected sothat the output signal at all times represents r-ro Thus, as soon as thetarget is detected passing through a known range dilference pointconstituting the factor Ad relay Rrx of the present disclosure isactuated so that the output signal is directly equal to tx-to wheretx-to is equal to At.

In this manner the invention provides a simple means of continuouslygenerating a variable signal representing the initial Velocity of atarget according to a hyperbolic function so that the instant the targetpasses through a second known range point an accurate signal may be readout.

In each of the above-ident-iied applications, it will be noted that thecapacitor C1 is also employed as a feedback device for achopper-stabilizer D.C. amplifier. In this manner, not only is theinitial velocity computation erformed in :accordance with the desiredhyperbolic function provided by the present invention, but also thefinal signal developed across capacitor C1 is incorporated into theintegrating feedback path as an initial condition for a subsequentintegration to form a varying velocity signal.

The equivalent circuit for the embodiment of Fig. 2 is shown in Fig. 2b,where it is assumed that the relay Ro is actuated, initiating thehyperbolic function generating operation, and that relay Rrx #has notbeen actuated.

The transform for the voltage ecl 4across C1 may then be expressed asfollows:

` 7 (eel) :E tran-k, kras-162]? Performing thezinverse trans-form thegenepaliform of" time functionvisfound to b e:

Suitable circuit constants their .bedeterrmined "byl comparing thesimulatingfunction desired withv this general time function; forexample:

l l l 1 l I-t will be noted that, since only three equations areprovided :and there are four unknowns in terms of circuit elements, thesolution which -is satisfactory is not unique. Therefore, `althoughprecise solutions may be determined mathematical-1y, it is consideredsatisfactory for the purpose of the invention to establish the basicconguran'on of elements and to approximate reasonable circuit valuestherefor, precise values being then determined experi satisfactory'forproviding the simulating ,exponentialy functions above: R1=6.8 megohmsC1`=2 microfarads R2'=6.8 megohms' C2=4 microfarads The generaltechnique of derivation in accordance with fthe present invention maynow be summarized as follows.

(1) The Afunction to be simulated 'as a continuous time function isapproximated by series of exponential which 'may be expressed 'asfollows:

(2) A plurality of sets of linear elements are then provided, one setbeing included for each of the terms in the exponential series.

(3) The. basic circuit configuration is then transformed in terms ofunknown element values to provide the transformation function:

A U(s) AL(s) where T(Eo) represents the transformed output signal, AU(s)represents the upper term in the transformation function and AL(s)represents the lower term, or root function which is equated to zero todetermine the roots k1' and k2, etc. In the example above:

(5) Finally the values of the various circuit constants are obtained byequating the unknown coetlicients A and B, etc., to the correspondingterms derived from the transformation providing the relationship:

A Uta) TUT) From the foregoing description, it is apparent that thepresent invention provides ya class of functioning generating circuitsrequiring only linear elements, Where these elements may :also be of apassive nature Vsuch as inductors, capacitors and resistors.

While the invention has been speciically illustrated with respect to aresistor-capacitor exponential circuit, it will be understood that otherexponential circuits may be utilized. Furthermore, it may be understoodthat, while the specific example illustrated relates to simulation of ahyperbola, other functions such as parabolas and ellipses may besimulated.

What is claimed is:

A hyperbolic function generator comprising: a iirst circuit includingdirect-current source of potential, a switch, and a resistor connectedin series; a second circuit including a first capacitor connected inparallel with said resistor; a third circuit including a resistor and asecond capacitor connected in series, said third circuit also beingconnected in parallel with said resistor; a switch connected in parallelwith said second capacitor; and means for opening said switchessimultaneously.

References Cited in the tile of this patent UNITED STATES PATENTS OTHERREFERENCES The Radio Amateurs Handbook, 1951 edition, published by theAmerican Radio Relay League, West Hartford, Conn.` Page 247 relied on.

Electronic Analogue Integration and Differentiation (Brandon),Electronic Engineering, November 1953, pages 476 and 477.

